Lattice Specific Heats near 0°K with an Application to Germanium

Formulas for the low-temperature lattice specific heat are developed on the basis of the general adiabatic and harmonic assumptions, independently of special models or numerical procedures. Explicit simple formulas are obtained for ${\theta }_D\mathrm{}\left(0\right)\mathrm{}$, the equivalent Debye characteristic temperature at 0°K, and for the curvature of ${\theta }_D\mathrm{}\left(T\right)\mathrm{}$ at 0°K. Discussions are given of the resulting dependence of ${\theta }_D\mathrm{}\left(0\right)\mathrm{}$ on physical parameters and the significance of the formula for ${\theta }_D\mathrm{}\left(0\right)\mathrm{}$ as a check on the basic assumptions, of the absence of a linear term in ${\theta }_D\mathrm{}\left(T\right)\mathrm{}$, and of the dependence of the curvature on the dispersion of elastic waves. ${\theta }_D\mathrm{}\left(0\right)\mathrm{}$ is calculated for Ge as 374.0°K; an error of ±2°K is estimated as due to errors in the elastic constants whereas the computational error is negligible. ${\theta }_D\mathrm{}\left(T\right)\mathrm{}$ is calculated for Ge for $\left[\frac{T}{{\theta }_D\mathrm{}\left(0\right)\mathrm{}}\right]<\mathrm{}0.11\mathrm{}$ using two models. The first is a simple model of the frequency spectrum which gives results like typical force-constant models, and disagrees with measurement. The second is a model of the frequency spectrum based on the direct measurements by inelastic neutron scattering; this model shows much greater dispersion, and gives much better agreement of ${\theta }_D\mathrm{}\left(T\right)\mathrm{}$ with measurement.